where
$$ Y(t) = [y_1(t),dots,y_m(t)]^T, $$
$$beta(t) = [beta_1(t),dots,beta_f(t)]^T,$$
$$ X: m times f$$
for a given time $t$. The design matrix $X$ defines the relation between the functions $beta$ and observations $y$. In general, the rank of $X$ should match the number of functions $f$. The FANOVA model can then be described by a specific form of $X$ such that